The comparable case is with Slow. If you are Slowed in the middle of your move, and you've already moved at least 2, your move stops.
Likewise with Daze, if you get dazed on your turn and you've already taken 1+ actions, you're done.
I'd argue it works the other way just as well!
-O

(Dungeons & Dragons)
Rulebook featuring "high magic" options, including a host of new spells.